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lecture8_web - L8-1EEL 6550 Error-control Coding Lecture...

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Unformatted text preview: L8-1EEL 6550 Error-control Coding Lecture 8•FieldsDEFNAfieldis a commutative ring with identity in which every element has aninverse under·.–Essentially, a field is:*a set of elementsF*with two binary operations+(addition)and·(multiplication)*“+”,“·”, andinversescan be used to doaddition, subtraction, multiplication, anddivisionwithout leaving the setDEFNFormal definition:Afieldconsists of a setFand two binary operations+and·that satisfy the following properties:1.Fforms acommutative group under addition (+).The additive identity is labeled “0”.2.F- {}forms acommutative group under multiplication(·).Themultiplicative identity is labeled “1”.3. The operation“.”distributes over+:a·(b+c) = (a·b) + (a·c).–Examples of Infinite Fields*The rational numbers*The integers do not form a field because they do not form a group under “·”. (Thereare no multiplicative inverses.)*The real numbers*The complex numbers•Finite Fields–Finite fieldsare more commonly known asGalois Fieldsafter their discoverer–AGalois fieldwithpmembers is denotedGF(p)–Every field must have at least 2 elements:*the additive identity ’0’, and*the multiplicative identity ’1’L8-2•There exists a finite field with 2 elements: thebinary field, denotedGF(2)–F={,1}–+defined as modulo-2 addition+1111–·defined as modulo-2 multiplication·111–It is easy to verify that·distributes of+by trying each of the 8 possible combinations•Given a prime numberp, the integers{,1,2, . . . , p-1}form a field under modulopaddition and multiplication....
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